Recent innovations in higher category theory have unlocked the potential to reimagine the basic tools and constructions in algebraic topology. This workshop will explore the interplay between these higher and $infty$-categorical techniques with classical algebraic topology, playing each off of the other and returning the field to conceptual, geometrical intuition.

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Algebraic geometry of topological field theories
David Ben-Zvi (University of Texas)

Location

MSRI: Simons Auditorium

Video

Abstract

The study of supersymmetric gauge theories in physics has produced an incredibly rich collection of mathematical structures (for example the construction of Seiberg-Witten integrable systems and their quantization). I will explain (following work in progress with D. Nadler, A. Neitzke and T. Nevins) how to see such structures directly from the formalism of extended topological field theory. The main (and utterly conjectural) example is provided by a 6-dimensional field theory, Theory X (better known as the (2,0) 6d SCFT), whose structure unifies much of modern geometric representation theory.

What is Poincaré duality for factorization homology? Our answer has three ingredients: Koszul duality, zero-pointed manifolds, and Goodwillie calculus. We introduce zero-pointed manifolds so as to construct a Poincaré duality map from factorization homology to factorization cohomology; this cohomology theory has coefficients the Koszul dual coalgebra. Goodwillie calculus is used to prove this Poincaré/Koszul duality when the coefficient algebra is connected. The key technical step is that Goodwillie calculus is Koszul dual to Goodwillie-Weiss calculus.

For g a dgla over a field of characteristic zero, the dual of the Hochschild homology of the universal enveloping algebra of g *completes* to the Hochschild homology of the Lie algebra cohomology of g. In this talk we will resolve this completion discrepancy through considerations of formal algebraic geometry. This will be an instance of our main result, which is a version of Poincare' duality for factorization homology as it interacts with Koszul duality in the sense of formal moduli. This can be interpreted as a duality among certain topological field theories that exchanges perturbative and non-perturbative.

One can phrase interesting objects in terms of fixed points of group actions. For example, class numbers of quadratic extensions of Q can be expressed with fixed points of actions on modular curves. Derived functors are frequently better behaved than their non-derived versions, so it is useful to consider the associated derived functor, called the homotopy fixed points. Sullivan's conjecture is an equivalence between appropriately completed spaces of fixed points and homotopy fixed points for finite p-groups. It was proven independently by H. Miller, G. Carlsson, and J. Lannes. This talk will present Sullivan's conjecture and its solutions, and discuss analogues for absolute Galois groups conjectured by Grothendieck.

A differential cohomology theory is differential geometric refinement of a generalized cohomology theory (in the sense of algebraic topology). Examples naturally arise in physics or in the study of secondary invariants (e.g. Chern-Simons invariants). We discuss this notion from a higher categorical point of view. This leads to a natural decomposition of any differential cohomology theory which we illustrate with many examples. Moreover we show how to obtain a good integration theory and a notion of twisted differential cohomology and discuss some aspects and examples.

Higher categories are playing an increasingly important role in algebraic topology and mathematics more generally. Due to their diverse origins there are many competing approaches to the theory. In this talk I will describe joint work with Clark Barwick which gives a solution to the comparison problem in higher category theory. We give a brief axiomatization of the theory of (infty,n)-categories (and other closely related theories). From this we show that the space of homotopy theories satisfying these axioms is B(Z/2)^n, and hence any two theories satisfying the axioms are equivalent with very little ambiguity in _how_ they are equivalent. Examples of popular theories which satisfy these axioms will be provided along with a spattering of applications.

Recent work has shown that many classical results do not carry through to the equivariant setting. Even notions like the Zariski topology are no longer right, since there are "degrees of commutativity" in equivariant stable homotopy theory that have no classical antecedents. This talk with describe what we know and make some guesses and observations about what we just don't understand.

I will talk about homotopical dualities of Brown-Comenetz type through the lens of algebra and algebraic geometry. In particular, I will discuss details of self-duality in the prominent examples of K-theory and topological modular forms.

I will describe two calculations, obtained using the theory power options for Morava E-theory at height 2: (1) The E- theory of the Bousfield-Kuhn spectrum (joint work with Mark Behrens) and (2) Twists of E-cohomology.

Let $R$ be an $E_{2}$ ring spectrum with zero odd dimensional homotopy groups. Every map of ring spectra $MU\to R$ is represented by a map of $E_{2}$ ring spectra. If $2$ is invertible in $\pi_{0}R$, then every map of ring spectra $MSO\to R$ is represented by a map of $E_{2}$ ring spectra. Joint work with Greg Chadwick.

Topological spaces and simplicial sets have associated homotopy types (that is, the 1-category of topological spaces maps to the infinity-category of spaces, and so does the 1-category of simplicial sets), but these are not the only kind of mathematical objects that have associated homotopy types. In this talk, I will present a mathematical object (not a topological

space) that comes from the theory of von Neumann algebras, and whose associated homotopy type is K(Z,4).